In the case of linear maps,
(REF ) corresponds to a “banded” estimator for the Cholesky factor of the precision matrix of the ensemble
\(z^1,\ldots ,z^M\) 
[1]}, [2]}, and has been used for the purpose of regularized covariance estimation in the context of data assimilation [3]}.
Indeed, if \(\pi = \mathcal {N}( {\bf 0}, \Sigma )\)  and \(\Sigma = LL^\top \)  is a Cholesky decomposition of
\(\Sigma \succ 0\) , then
\(SL^{-1}\) 
represents a linear KR rearrangement
that pushes forward \(\pi \)  to
a standard normal.
In this paper, we extend these banded estimators to the
nonlinear/non-Gaussian case, using the
KR rearrangement to generalize
the notion of a Cholesky factor.
